Maclaurin series is a special case of Taylor series when c is equal to zero.
With Fourier series, we are interested in expanding a function as an infinite series of sines and cosines. Other functions can also be used instead of sines and cosines like the Legendre polynomials for more general expansions but we will be using sines and cosines for the examples. A Fourier series expansion is written as,

f(x)=a_0+\sum_{n=1}^{\infty}\left (a_n\cos{nx}+b_n\sin{nx}\right )

where a_0, a_n, and b_n for n =1, 2, 3,..., are called the Fourier coefficients.

Deriving the Coefficients for 2π-periodic function

Suppose that the 2π-periodic function f has the Fourier series representation

f(x)=a_0+\sum_{n=1}^{\infty}\left (a_n\cos{nx}+b_n\sin{nx}\right )

The coefficients can then be solved using Euler formulas and are illustrated below.

Computing a_0

From Equation (2), we integrate both sides over the interval [-\pi, \pi].